- Start with the given formula \(P = 2(l + b)\).
- Divide both sides by \(2\) to obtain \(\frac{P}{2} = l + b\).
- Subtract \(b\) from both sides to isolate \(l\), yielding \(l = \frac{P}{2} - b\).
- Combine the terms over a common denominator to reach the final form \(\boxed{l = \frac{P - 2b}{2}}\).
Explanation
- Problem Analysis
We are given the perimeter formula for a rectangle: \(P = 2(l + b)\), where \(P\) is the perimeter, \(l\) is the length, and \(b\) is the width. Our objective is to isolate the variable \(l\) to express it in terms of \(P\) and \(b\). This involves standard algebraic manipulation steps, such as division and subtraction, to transform the equation into the desired form.
- Algebraic Manipulation
Starting with the equation \(P = 2(l + b)\), we first divide both sides by 2 to isolate the expression \((l + b)\):
\[\frac{P}{2} = l + b\]
Next, to solve for \(l\), we subtract \(b\) from both sides of the equation:
\[l = \frac{P}{2} - b\]
To combine these terms into a single fraction, we find a common denominator, which is 2:
\[l = \frac{P}{2} - \frac{2b}{2} = \frac{P - 2b}{2}\]
Thus, the formula solved for \(l\) is \(l = \frac{P - 2b}{2}\).
- Final Conclusion
Comparing our derived result \(l = \frac{P - 2b}{2}\) with the given options, we find that it matches the second option. Therefore, the correct expression for \(l\) is \(\boxed{l = \frac{P - 2b}{2}}\).
Examples
In real-life scenarios, architects and construction planners often use the perimeter formula to determine the dimensions of rooms or plots of land. If an architect knows the total amount of fencing (the perimeter \(P\)) required for a rectangular garden and the width \(b\) of the plot, they can quickly determine the necessary length \(l\) by rearranging the formula into \(l = \frac{P - 2b}{2}\). This allows for efficient planning and resource management in design projects.